It was stated by E. Lindelöf [1]. The Lindelöf conjecture is equivalent to the assertion that for a fixed $\sigma\in(1/2,1)$ the number of zeros of $\zeta(s)$ that lie in the domain $\operatorname{Re}s>\sigma,T<\operatorname{Im}s<T+1$ is $o(\ln T)$. The Lindelöf conjecture is therefore a consequence of the Riemann conjecture on the zeros of $\zeta(s)$ (cf. Riemann hypotheses). It is known (1982) that

$$\varlimsup_{t\to\infty}\frac{|\zeta(1/2+it)|}{t^c}=0,$$

where $c$ is a constant such that $0<c<6/37$.

There is a generalization of the Lindelöf conjecture to Dirichlet $L$-functions: For any $\epsilon>0$,